x NOTATIO N GUID E GL(2, C) Invertibl e 2 x 2 matrice s wit h comple x 2 2 entries. S{w) Grap h o f w e M mxn = { (wzz): z E C n } . 2 3 Ay Th e dis k i n M m x n give n b y 2 5 {w: w*w -f f*w + w*f + d 0} where F i s the selfadjoin t matri x ( ?m £) . [ , ] y Sesquilinea r for m o n C 2 define d b y 2 5 [u,v]Y = [yu,v] . K [ , ] Krei n space , a Hilbert spac e K wit h a 2 5 nondegenerate sesquilinea r for m [ , ] . K + , K _ Hilber t orthogona l subspace s whos e direc t 2 5 sum i s K, s o that [, ] = [ , ]y , Y = ( Q _°7) with respec t t o thi s decomposition . M' Close d [ , ]-orthogona l complemen t o f the 2 5 subspace M . M - f N Su m o f M an d N i f they ar e disjoint , 2 6 ( , ]-orthogona l an d i f M - f N i s closed . C(wi,W2,W3,W4) Cros s rati o o f the comple x number s 2 9 C(wi,W2,W3,W4) = (Wi - W2){W 2 - W3)" 1 (W 3 - W 4 ){W4 - Wi)' 1 . Chapter 4 Vector an d matrix-value d analog s o f L p , Hp 3 6 on th e uni t disk . Vector an d matrix-value d analog s o f 3 6 L oo ^0 0 o n th e uni t disk . All function s i n L P N whos e positiv e Fourie r 3 6 coefficients vanish . Toeplitz operator . 3 7 Annihilator o f M i n L q N . 4 1 Chapter 5 Tl Th e se t { / e BH°° l : f{z 3 ) = w 3 , 4 4 J = l , . . . , p } . clos T1 Th e closur e o f the se t T ' i n L°° . 4 4 A { 2 y W } Th e "Pick " matri x { i = ^ } ^ = = 1 4 5 /c^(e^) Th e Szeg o reproducing kerne l ^ 1 i9 . 4 5 C^ Th e se t {(^ ) eBH^: Kp = a x hx + a2h2 4 7 is rational, ha s \t{e %e )\ = 1 , has windin g number abou t zer o I}. Her e a\,a 2 H°°, K 0, and / an intege r ar e given . TV ro o Hr N TA M1 ^ M x r o o N' :N ' - " i V ' rroo HPMx N rroo
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